// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Keir Mierle <mierle@gmail.com>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2011 Timothy E. Holy <tim.holy@gmail.com >
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_LDLT_H
#define EIGEN_LDLT_H

namespace Eigen {

namespace internal {
template<typename _MatrixType, int _UpLo>
struct traits<LDLT<_MatrixType, _UpLo>> : traits<_MatrixType>
{
	typedef MatrixXpr XprKind;
	typedef SolverStorage StorageKind;
	typedef int StorageIndex;
	enum
	{
		Flags = 0
	};
};

template<typename MatrixType, int UpLo>
struct LDLT_Traits;

// PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef
enum SignMatrix
{
	PositiveSemiDef,
	NegativeSemiDef,
	ZeroSign,
	Indefinite
};
}

/** \ingroup Cholesky_Module
 *
 * \class LDLT
 *
 * \brief Robust Cholesky decomposition of a matrix with pivoting
 *
 * \tparam _MatrixType the type of the matrix of which to compute the LDL^T Cholesky decomposition
 * \tparam _UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
 *             The other triangular part won't be read.
 *
 * Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite
 * matrix \f$ A \f$ such that \f$ A =  P^TLDL^*P \f$, where P is a permutation matrix, L
 * is lower triangular with a unit diagonal and D is a diagonal matrix.
 *
 * The decomposition uses pivoting to ensure stability, so that D will have
 * zeros in the bottom right rank(A) - n submatrix. Avoiding the square root
 * on D also stabilizes the computation.
 *
 * Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky
 * decomposition to determine whether a system of equations has a solution.
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::ldlt(), SelfAdjointView::ldlt(), class LLT
 */
template<typename _MatrixType, int _UpLo>
class LDLT : public SolverBase<LDLT<_MatrixType, _UpLo>>
{
  public:
	typedef _MatrixType MatrixType;
	typedef SolverBase<LDLT> Base;
	friend class SolverBase<LDLT>;

	EIGEN_GENERIC_PUBLIC_INTERFACE(LDLT)
	enum
	{
		MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
		MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
		UpLo = _UpLo
	};
	typedef Matrix<Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1> TmpMatrixType;

	typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
	typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;

	typedef internal::LDLT_Traits<MatrixType, UpLo> Traits;

	/** \brief Default Constructor.
	 *
	 * The default constructor is useful in cases in which the user intends to
	 * perform decompositions via LDLT::compute(const MatrixType&).
	 */
	LDLT()
		: m_matrix()
		, m_transpositions()
		, m_sign(internal::ZeroSign)
		, m_isInitialized(false)
	{
	}

	/** \brief Default Constructor with memory preallocation
	 *
	 * Like the default constructor but with preallocation of the internal data
	 * according to the specified problem \a size.
	 * \sa LDLT()
	 */
	explicit LDLT(Index size)
		: m_matrix(size, size)
		, m_transpositions(size)
		, m_temporary(size)
		, m_sign(internal::ZeroSign)
		, m_isInitialized(false)
	{
	}

	/** \brief Constructor with decomposition
	 *
	 * This calculates the decomposition for the input \a matrix.
	 *
	 * \sa LDLT(Index size)
	 */
	template<typename InputType>
	explicit LDLT(const EigenBase<InputType>& matrix)
		: m_matrix(matrix.rows(), matrix.cols())
		, m_transpositions(matrix.rows())
		, m_temporary(matrix.rows())
		, m_sign(internal::ZeroSign)
		, m_isInitialized(false)
	{
		compute(matrix.derived());
	}

	/** \brief Constructs a LDLT factorization from a given matrix
	 *
	 * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c
	 * MatrixType is a Eigen::Ref.
	 *
	 * \sa LDLT(const EigenBase&)
	 */
	template<typename InputType>
	explicit LDLT(EigenBase<InputType>& matrix)
		: m_matrix(matrix.derived())
		, m_transpositions(matrix.rows())
		, m_temporary(matrix.rows())
		, m_sign(internal::ZeroSign)
		, m_isInitialized(false)
	{
		compute(matrix.derived());
	}

	/** Clear any existing decomposition
	 * \sa rankUpdate(w,sigma)
	 */
	void setZero() { m_isInitialized = false; }

	/** \returns a view of the upper triangular matrix U */
	inline typename Traits::MatrixU matrixU() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return Traits::getU(m_matrix);
	}

	/** \returns a view of the lower triangular matrix L */
	inline typename Traits::MatrixL matrixL() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return Traits::getL(m_matrix);
	}

	/** \returns the permutation matrix P as a transposition sequence.
	 */
	inline const TranspositionType& transpositionsP() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return m_transpositions;
	}

	/** \returns the coefficients of the diagonal matrix D */
	inline Diagonal<const MatrixType> vectorD() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return m_matrix.diagonal();
	}

	/** \returns true if the matrix is positive (semidefinite) */
	inline bool isPositive() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
	}

	/** \returns true if the matrix is negative (semidefinite) */
	inline bool isNegative(void) const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
	}

#ifdef EIGEN_PARSED_BY_DOXYGEN
	/** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A.
	 *
	 * This function also supports in-place solves using the syntax <tt>x = decompositionObject.solve(x)</tt> .
	 *
	 * \note_about_checking_solutions
	 *
	 * More precisely, this method solves \f$ A x = b \f$ using the decomposition \f$ A = P^T L D L^* P \f$
	 * by solving the systems \f$ P^T y_1 = b \f$, \f$ L y_2 = y_1 \f$, \f$ D y_3 = y_2 \f$,
	 * \f$ L^* y_4 = y_3 \f$ and \f$ P x = y_4 \f$ in succession. If the matrix \f$ A \f$ is singular, then
	 * \f$ D \f$ will also be singular (all the other matrices are invertible). In that case, the
	 * least-square solution of \f$ D y_3 = y_2 \f$ is computed. This does not mean that this function
	 * computes the least-square solution of \f$ A x = b \f$ if \f$ A \f$ is singular.
	 *
	 * \sa MatrixBase::ldlt(), SelfAdjointView::ldlt()
	 */
	template<typename Rhs>
	inline const Solve<LDLT, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

	template<typename Derived>
	bool solveInPlace(MatrixBase<Derived>& bAndX) const;

	template<typename InputType>
	LDLT& compute(const EigenBase<InputType>& matrix);

	/** \returns an estimate of the reciprocal condition number of the matrix of
	 *  which \c *this is the LDLT decomposition.
	 */
	RealScalar rcond() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return internal::rcond_estimate_helper(m_l1_norm, *this);
	}

	template<typename Derived>
	LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha = 1);

	/** \returns the internal LDLT decomposition matrix
	 *
	 * TODO: document the storage layout
	 */
	inline const MatrixType& matrixLDLT() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return m_matrix;
	}

	MatrixType reconstructedMatrix() const;

	/** \returns the adjoint of \c *this, that is, a const reference to the decomposition itself as the underlying
	 * matrix is self-adjoint.
	 *
	 * This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:
	 * \code x = decomposition.adjoint().solve(b) \endcode
	 */
	const LDLT& adjoint() const { return *this; };

	EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows(); }
	EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }

	/** \brief Reports whether previous computation was successful.
	 *
	 * \returns \c Success if computation was successful,
	 *          \c NumericalIssue if the factorization failed because of a zero pivot.
	 */
	ComputationInfo info() const
	{
		eigen_assert(m_isInitialized && "LDLT is not initialized.");
		return m_info;
	}

#ifndef EIGEN_PARSED_BY_DOXYGEN
	template<typename RhsType, typename DstType>
	void _solve_impl(const RhsType& rhs, DstType& dst) const;

	template<bool Conjugate, typename RhsType, typename DstType>
	void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

  protected:
	static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

	/** \internal
	 * Used to compute and store the Cholesky decomposition A = L D L^* = U^* D U.
	 * The strict upper part is used during the decomposition, the strict lower
	 * part correspond to the coefficients of L (its diagonal is equal to 1 and
	 * is not stored), and the diagonal entries correspond to D.
	 */
	MatrixType m_matrix;
	RealScalar m_l1_norm;
	TranspositionType m_transpositions;
	TmpMatrixType m_temporary;
	internal::SignMatrix m_sign;
	bool m_isInitialized;
	ComputationInfo m_info;
};

namespace internal {

template<int UpLo>
struct ldlt_inplace;

template<>
struct ldlt_inplace<Lower>
{
	template<typename MatrixType, typename TranspositionType, typename Workspace>
	static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
	{
		using std::abs;
		typedef typename MatrixType::Scalar Scalar;
		typedef typename MatrixType::RealScalar RealScalar;
		typedef typename TranspositionType::StorageIndex IndexType;
		eigen_assert(mat.rows() == mat.cols());
		const Index size = mat.rows();
		bool found_zero_pivot = false;
		bool ret = true;

		if (size <= 1) {
			transpositions.setIdentity();
			if (size == 0)
				sign = ZeroSign;
			else if (numext::real(mat.coeff(0, 0)) > static_cast<RealScalar>(0))
				sign = PositiveSemiDef;
			else if (numext::real(mat.coeff(0, 0)) < static_cast<RealScalar>(0))
				sign = NegativeSemiDef;
			else
				sign = ZeroSign;
			return true;
		}

		for (Index k = 0; k < size; ++k) {
			// Find largest diagonal element
			Index index_of_biggest_in_corner;
			mat.diagonal().tail(size - k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
			index_of_biggest_in_corner += k;

			transpositions.coeffRef(k) = IndexType(index_of_biggest_in_corner);
			if (k != index_of_biggest_in_corner) {
				// apply the transposition while taking care to consider only
				// the lower triangular part
				Index s = size - index_of_biggest_in_corner - 1; // trailing size after the biggest element
				mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
				mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
				std::swap(mat.coeffRef(k, k), mat.coeffRef(index_of_biggest_in_corner, index_of_biggest_in_corner));
				for (Index i = k + 1; i < index_of_biggest_in_corner; ++i) {
					Scalar tmp = mat.coeffRef(i, k);
					mat.coeffRef(i, k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner, i));
					mat.coeffRef(index_of_biggest_in_corner, i) = numext::conj(tmp);
				}
				if (NumTraits<Scalar>::IsComplex)
					mat.coeffRef(index_of_biggest_in_corner, k) =
						numext::conj(mat.coeff(index_of_biggest_in_corner, k));
			}

			// partition the matrix:
			//       A00 |  -  |  -
			// lu  = A10 | A11 |  -
			//       A20 | A21 | A22
			Index rs = size - k - 1;
			Block<MatrixType, Dynamic, 1> A21(mat, k + 1, k, rs, 1);
			Block<MatrixType, 1, Dynamic> A10(mat, k, 0, 1, k);
			Block<MatrixType, Dynamic, Dynamic> A20(mat, k + 1, 0, rs, k);

			if (k > 0) {
				temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
				mat.coeffRef(k, k) -= (A10 * temp.head(k)).value();
				if (rs > 0)
					A21.noalias() -= A20 * temp.head(k);
			}

			// In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
			// was smaller than the cutoff value. However, since LDLT is not rank-revealing
			// we should only make sure that we do not introduce INF or NaN values.
			// Remark that LAPACK also uses 0 as the cutoff value.
			RealScalar realAkk = numext::real(mat.coeffRef(k, k));
			bool pivot_is_valid = (abs(realAkk) > RealScalar(0));

			if (k == 0 && !pivot_is_valid) {
				// The entire diagonal is zero, there is nothing more to do
				// except filling the transpositions, and checking whether the matrix is zero.
				sign = ZeroSign;
				for (Index j = 0; j < size; ++j) {
					transpositions.coeffRef(j) = IndexType(j);
					ret = ret && (mat.col(j).tail(size - j - 1).array() == Scalar(0)).all();
				}
				return ret;
			}

			if ((rs > 0) && pivot_is_valid)
				A21 /= realAkk;
			else if (rs > 0)
				ret = ret && (A21.array() == Scalar(0)).all();

			if (found_zero_pivot && pivot_is_valid)
				ret = false; // factorization failed
			else if (!pivot_is_valid)
				found_zero_pivot = true;

			if (sign == PositiveSemiDef) {
				if (realAkk < static_cast<RealScalar>(0))
					sign = Indefinite;
			} else if (sign == NegativeSemiDef) {
				if (realAkk > static_cast<RealScalar>(0))
					sign = Indefinite;
			} else if (sign == ZeroSign) {
				if (realAkk > static_cast<RealScalar>(0))
					sign = PositiveSemiDef;
				else if (realAkk < static_cast<RealScalar>(0))
					sign = NegativeSemiDef;
			}
		}

		return ret;
	}

	// Reference for the algorithm: Davis and Hager, "Multiple Rank
	// Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
	// Trivial rearrangements of their computations (Timothy E. Holy)
	// allow their algorithm to work for rank-1 updates even if the
	// original matrix is not of full rank.
	// Here only rank-1 updates are implemented, to reduce the
	// requirement for intermediate storage and improve accuracy
	template<typename MatrixType, typename WDerived>
	static bool updateInPlace(MatrixType& mat,
							  MatrixBase<WDerived>& w,
							  const typename MatrixType::RealScalar& sigma = 1)
	{
		using numext::isfinite;
		typedef typename MatrixType::Scalar Scalar;
		typedef typename MatrixType::RealScalar RealScalar;

		const Index size = mat.rows();
		eigen_assert(mat.cols() == size && w.size() == size);

		RealScalar alpha = 1;

		// Apply the update
		for (Index j = 0; j < size; j++) {
			// Check for termination due to an original decomposition of low-rank
			if (!(isfinite)(alpha))
				break;

			// Update the diagonal terms
			RealScalar dj = numext::real(mat.coeff(j, j));
			Scalar wj = w.coeff(j);
			RealScalar swj2 = sigma * numext::abs2(wj);
			RealScalar gamma = dj * alpha + swj2;

			mat.coeffRef(j, j) += swj2 / alpha;
			alpha += swj2 / dj;

			// Update the terms of L
			Index rs = size - j - 1;
			w.tail(rs) -= wj * mat.col(j).tail(rs);
			if (gamma != 0)
				mat.col(j).tail(rs) += (sigma * numext::conj(wj) / gamma) * w.tail(rs);
		}
		return true;
	}

	template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
	static bool update(MatrixType& mat,
					   const TranspositionType& transpositions,
					   Workspace& tmp,
					   const WType& w,
					   const typename MatrixType::RealScalar& sigma = 1)
	{
		// Apply the permutation to the input w
		tmp = transpositions * w;

		return ldlt_inplace<Lower>::updateInPlace(mat, tmp, sigma);
	}
};

template<>
struct ldlt_inplace<Upper>
{
	template<typename MatrixType, typename TranspositionType, typename Workspace>
	static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat,
											  TranspositionType& transpositions,
											  Workspace& temp,
											  SignMatrix& sign)
	{
		Transpose<MatrixType> matt(mat);
		return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
	}

	template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
	static EIGEN_STRONG_INLINE bool update(MatrixType& mat,
										   TranspositionType& transpositions,
										   Workspace& tmp,
										   WType& w,
										   const typename MatrixType::RealScalar& sigma = 1)
	{
		Transpose<MatrixType> matt(mat);
		return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
	}
};

template<typename MatrixType>
struct LDLT_Traits<MatrixType, Lower>
{
	typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
	typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
	static inline MatrixL getL(const MatrixType& m) { return MatrixL(m); }
	static inline MatrixU getU(const MatrixType& m) { return MatrixU(m.adjoint()); }
};

template<typename MatrixType>
struct LDLT_Traits<MatrixType, Upper>
{
	typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
	typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
	static inline MatrixL getL(const MatrixType& m) { return MatrixL(m.adjoint()); }
	static inline MatrixU getU(const MatrixType& m) { return MatrixU(m); }
};

} // end namespace internal

/** Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of \a matrix
 */
template<typename MatrixType, int _UpLo>
template<typename InputType>
LDLT<MatrixType, _UpLo>&
LDLT<MatrixType, _UpLo>::compute(const EigenBase<InputType>& a)
{
	check_template_parameters();

	eigen_assert(a.rows() == a.cols());
	const Index size = a.rows();

	m_matrix = a.derived();

	// Compute matrix L1 norm = max abs column sum.
	m_l1_norm = RealScalar(0);
	// TODO move this code to SelfAdjointView
	for (Index col = 0; col < size; ++col) {
		RealScalar abs_col_sum;
		if (_UpLo == Lower)
			abs_col_sum = m_matrix.col(col).tail(size - col).template lpNorm<1>() +
						  m_matrix.row(col).head(col).template lpNorm<1>();
		else
			abs_col_sum = m_matrix.col(col).head(col).template lpNorm<1>() +
						  m_matrix.row(col).tail(size - col).template lpNorm<1>();
		if (abs_col_sum > m_l1_norm)
			m_l1_norm = abs_col_sum;
	}

	m_transpositions.resize(size);
	m_isInitialized = false;
	m_temporary.resize(size);
	m_sign = internal::ZeroSign;

	m_info = internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign) ? Success
																									  : NumericalIssue;

	m_isInitialized = true;
	return *this;
}

/** Update the LDLT decomposition:  given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.
 * \param w a vector to be incorporated into the decomposition.
 * \param sigma a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column
 * vectors. Optional; default value is +1. \sa setZero()
 */
template<typename MatrixType, int _UpLo>
template<typename Derived>
LDLT<MatrixType, _UpLo>&
LDLT<MatrixType, _UpLo>::rankUpdate(const MatrixBase<Derived>& w,
									const typename LDLT<MatrixType, _UpLo>::RealScalar& sigma)
{
	typedef typename TranspositionType::StorageIndex IndexType;
	const Index size = w.rows();
	if (m_isInitialized) {
		eigen_assert(m_matrix.rows() == size);
	} else {
		m_matrix.resize(size, size);
		m_matrix.setZero();
		m_transpositions.resize(size);
		for (Index i = 0; i < size; i++)
			m_transpositions.coeffRef(i) = IndexType(i);
		m_temporary.resize(size);
		m_sign = sigma >= 0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
		m_isInitialized = true;
	}

	internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);

	return *this;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template<typename _MatrixType, int _UpLo>
template<typename RhsType, typename DstType>
void
LDLT<_MatrixType, _UpLo>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
	_solve_impl_transposed<true>(rhs, dst);
}

template<typename _MatrixType, int _UpLo>
template<bool Conjugate, typename RhsType, typename DstType>
void
LDLT<_MatrixType, _UpLo>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
	// dst = P b
	dst = m_transpositions * rhs;

	// dst = L^-1 (P b)
	// dst = L^-*T (P b)
	matrixL().template conjugateIf<!Conjugate>().solveInPlace(dst);

	// dst = D^-* (L^-1 P b)
	// dst = D^-1 (L^-*T P b)
	// more precisely, use pseudo-inverse of D (see bug 241)
	using std::abs;
	const typename Diagonal<const MatrixType>::RealReturnType vecD(vectorD());
	// In some previous versions, tolerance was set to the max of 1/highest (or rather numeric_limits::min())
	// and the maximal diagonal entry * epsilon as motivated by LAPACK's xGELSS:
	// RealScalar tolerance = numext::maxi(vecD.array().abs().maxCoeff() *
	// NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest()); However, LDLT is not rank
	// revealing, and so adjusting the tolerance wrt to the highest diagonal element is not well justified and leads to
	// numerical issues in some cases. Moreover, Lapack's xSYTRS routines use 0 for the tolerance. Using
	// numeric_limits::min() gives us more robustness to denormals.
	RealScalar tolerance = (std::numeric_limits<RealScalar>::min)();
	for (Index i = 0; i < vecD.size(); ++i) {
		if (abs(vecD(i)) > tolerance)
			dst.row(i) /= vecD(i);
		else
			dst.row(i).setZero();
	}

	// dst = L^-* (D^-* L^-1 P b)
	// dst = L^-T (D^-1 L^-*T P b)
	matrixL().transpose().template conjugateIf<Conjugate>().solveInPlace(dst);

	// dst = P^T (L^-* D^-* L^-1 P b) = A^-1 b
	// dst = P^-T (L^-T D^-1 L^-*T P b) = A^-1 b
	dst = m_transpositions.transpose() * dst;
}
#endif

/** \internal use x = ldlt_object.solve(x);
 *
 * This is the \em in-place version of solve().
 *
 * \param bAndX represents both the right-hand side matrix b and result x.
 *
 * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
 *
 * This version avoids a copy when the right hand side matrix b is not
 * needed anymore.
 *
 * \sa LDLT::solve(), MatrixBase::ldlt()
 */
template<typename MatrixType, int _UpLo>
template<typename Derived>
bool
LDLT<MatrixType, _UpLo>::solveInPlace(MatrixBase<Derived>& bAndX) const
{
	eigen_assert(m_isInitialized && "LDLT is not initialized.");
	eigen_assert(m_matrix.rows() == bAndX.rows());

	bAndX = this->solve(bAndX);

	return true;
}

/** \returns the matrix represented by the decomposition,
 * i.e., it returns the product: P^T L D L^* P.
 * This function is provided for debug purpose. */
template<typename MatrixType, int _UpLo>
MatrixType
LDLT<MatrixType, _UpLo>::reconstructedMatrix() const
{
	eigen_assert(m_isInitialized && "LDLT is not initialized.");
	const Index size = m_matrix.rows();
	MatrixType res(size, size);

	// P
	res.setIdentity();
	res = transpositionsP() * res;
	// L^* P
	res = matrixU() * res;
	// D(L^*P)
	res = vectorD().real().asDiagonal() * res;
	// L(DL^*P)
	res = matrixL() * res;
	// P^T (LDL^*P)
	res = transpositionsP().transpose() * res;

	return res;
}

/** \cholesky_module
 * \returns the Cholesky decomposition with full pivoting without square root of \c *this
 * \sa MatrixBase::ldlt()
 */
template<typename MatrixType, unsigned int UpLo>
inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
SelfAdjointView<MatrixType, UpLo>::ldlt() const
{
	return LDLT<PlainObject, UpLo>(m_matrix);
}

/** \cholesky_module
 * \returns the Cholesky decomposition with full pivoting without square root of \c *this
 * \sa SelfAdjointView::ldlt()
 */
template<typename Derived>
inline const LDLT<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::ldlt() const
{
	return LDLT<PlainObject>(derived());
}

} // end namespace Eigen

#endif // EIGEN_LDLT_H
